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Properties

Label 8-8512e4-1.1-c1e4-0-3
Degree $8$
Conductor $5.250\times 10^{15}$
Sign $1$
Analytic cond. $2.13419\times 10^{7}$
Root an. cond. $8.24431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s − 4·7-s − 3·9-s + 2·11-s − 2·13-s − 2·15-s + 3·17-s + 4·19-s − 8·21-s − 6·23-s − 11·25-s − 8·27-s − 17·31-s + 4·33-s + 4·35-s − 3·37-s − 4·39-s + 8·41-s + 7·43-s + 3·45-s − 5·47-s + 10·49-s + 6·51-s + 16·53-s − 2·55-s + 8·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 1.51·7-s − 9-s + 0.603·11-s − 0.554·13-s − 0.516·15-s + 0.727·17-s + 0.917·19-s − 1.74·21-s − 1.25·23-s − 2.19·25-s − 1.53·27-s − 3.05·31-s + 0.696·33-s + 0.676·35-s − 0.493·37-s − 0.640·39-s + 1.24·41-s + 1.06·43-s + 0.447·45-s − 0.729·47-s + 10/7·49-s + 0.840·51-s + 2.19·53-s − 0.269·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 7^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2.13419\times 10^{7}\)
Root analytic conductor: \(8.24431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 7^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
19$C_1$ \( ( 1 - T )^{4} \)
good3$D_{4}$ \( ( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + T + 12 T^{2} + 6 T^{3} + 73 T^{4} + 6 p T^{5} + 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 17 T^{2} - 38 T^{3} + 211 T^{4} - 38 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 31 T^{2} + 56 T^{3} + 472 T^{4} + 56 p T^{5} + 31 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 129 T^{3} + 1220 T^{4} - 129 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 83 T^{2} + 360 T^{3} + 2724 T^{4} + 360 p T^{5} + 83 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 63 T^{2} - 78 T^{3} + 67 p T^{4} - 78 p T^{5} + 63 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 191 T^{2} + 1473 T^{3} + 9284 T^{4} + 1473 p T^{5} + 191 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 62 T^{2} + 264 T^{3} + 3393 T^{4} + 264 p T^{5} + 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 135 T^{2} - 882 T^{3} + 7639 T^{4} - 882 p T^{5} + 135 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 61 T^{2} - 223 T^{3} + 2704 T^{4} - 223 p T^{5} + 61 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 156 T^{2} + 546 T^{3} + 10159 T^{4} + 546 p T^{5} + 156 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 281 T^{2} - 2584 T^{3} + 24145 T^{4} - 2584 p T^{5} + 281 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 126 T^{2} - 1128 T^{3} + 8323 T^{4} - 1128 p T^{5} + 126 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - T + 128 T^{2} - 42 T^{3} + 10817 T^{4} - 42 p T^{5} + 128 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 107 T^{2} - 1059 T^{3} + 7892 T^{4} - 1059 p T^{5} + 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 27 T + 438 T^{2} + 4926 T^{3} + 45845 T^{4} + 4926 p T^{5} + 438 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + T + 191 T^{2} - 51 T^{3} + 17120 T^{4} - 51 p T^{5} + 191 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 245 T^{2} + 3015 T^{3} + 26640 T^{4} + 3015 p T^{5} + 245 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 143 T^{2} - 855 T^{3} + 11256 T^{4} - 855 p T^{5} + 143 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 311 T^{2} + 2241 T^{3} + 39840 T^{4} + 2241 p T^{5} + 311 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 176 T^{2} - 498 T^{3} + 23073 T^{4} - 498 p T^{5} + 176 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.92962179821772645926561977249, −5.48783170140982733158963367084, −5.45034528029403190039472433158, −5.42763846219249566817339070275, −5.32271993008993265830641406512, −5.01565320244563943735184310085, −4.56685208541873902729959778973, −4.43796485390945397055290015805, −4.20284665906695791154199002893, −3.94188310587351535642588659361, −3.90120421532800767736819123347, −3.73872239087431965837354194838, −3.66344985227290340121234091358, −3.32453245496974736050511290645, −3.25410657370744362198123907460, −3.01117117694639882474201372180, −2.80257290617079521098842732344, −2.39179393569249789419267673697, −2.36012772819937617231130364754, −2.33902756494717339511112375719, −2.17284018957063824146391491346, −1.56794415723388425181459391583, −1.36831559197853882792199188124, −1.15636910566227438850835245484, −0.995802341513401519648104055955, 0, 0, 0, 0, 0.995802341513401519648104055955, 1.15636910566227438850835245484, 1.36831559197853882792199188124, 1.56794415723388425181459391583, 2.17284018957063824146391491346, 2.33902756494717339511112375719, 2.36012772819937617231130364754, 2.39179393569249789419267673697, 2.80257290617079521098842732344, 3.01117117694639882474201372180, 3.25410657370744362198123907460, 3.32453245496974736050511290645, 3.66344985227290340121234091358, 3.73872239087431965837354194838, 3.90120421532800767736819123347, 3.94188310587351535642588659361, 4.20284665906695791154199002893, 4.43796485390945397055290015805, 4.56685208541873902729959778973, 5.01565320244563943735184310085, 5.32271993008993265830641406512, 5.42763846219249566817339070275, 5.45034528029403190039472433158, 5.48783170140982733158963367084, 5.92962179821772645926561977249

Graph of the $Z$-function along the critical line